Electric-wave filter



Jah. 29, 1935. w. cAuER ELECTRIC WAVE FILTER Filed Deo. 6, 1930. 4 Sheets-Shea?I l Jan. 29, 1935. w. CAUER 1,989,545

ELECTRIC WAVE FILTER Filed-Dec. 6, 1930 4 Sheets-Sheet 2 ATTORNEY Jan. 29, 1935. wF C'AUER I 1,989,545

ELECTRIC WAVE FILTER Filed Deo. 6, 1930 4 Sheet's-ShLm.v 5

A TToRNEY 1m29, 1935. w. CAUER 1,989,545

ELECTRIC wAvE FILTER y Filed Deo. 6, 1930 4 Sheets-Sheet 4 ffy/5 /N VENTO/2 Wilhelm Cauer BY y@ ATToRNEY.

Patented Jan. 29, 1935 PATE ko .FA-rflc'IE' Y ELECTRIC-WAVE FILTER Wilhelm Cauet, Gottingen, Germany Application December 6, 1930, Serial No. 500,595

In Germany June 8, 1928 73 Claims.

My invention relates to electric-wave lters, and more particularly to filters adapted for the selection and propagation of electromagnetic signals of prescribed ranges of frequency. From a more limited aspect, the invention relates to symmetrical wave filters, as well as to such unsymmetrical filters as are equivalent to a symmetrical lter connected in series with a transformer, or with a phase-correction circuit, or both.

One of the objects of this invention is to provide new and improved electric-wave filters, some of them of new types, never before known, and others, also new, but having certain features of equivalency with known filters.

A second object is to provide new and improved electric-wave-fllter circuits that shall have the property that the image impedance in the transmission bands shall be substantially constant to any required degree of accuracy.

A further object is to 'improve the behavior of the working vor total attenuation of filter networks.

Another object is to provide a new and improved wave lter that, when connected with a long line, shall reduce the echo or reflection effect to a value as small as may be desired.

It is further an object of my invention to improve upon lters of types already known, to the ends that their efliciency may be improved, their cost of manufacture lessened, and the number of elements, of which they; arecomposed, reduced.

Other and still further objects will be explained hereinafter, and will be particularly'pointed out in the appended claims, it being understood that I intend, by suitable expressions in the claims, to specify all the novelty that the invention may possess. ,y y

The invention will now be explained in connection withthe accompanying drawings, in which Fig. 1 is a diagrammatic view illustrating a fourterminal, symmetrical filter embodying the present invention, the filter being'shown in Wheatstone-bridge-network form; Fig. 2 is a similar view illustrating circuit elements that may be em.- bodied in the circuit connections of Fig. 1; Fig, 3 is a diagrammatic view of an unsymmetrical lter circuit arranged according to one general form of the present invention; Fig. 4 is a corresponding view of a symmetrical filter circuit according to my invention, this circuit also being of quite general form; Figs. 5 and 6 are diagrammatic views of prior-art filters for comparison purposes; Figs. 7 to 12 are diagrammatic views of filter circuits builtaccording to the present invention; Figs. 13 and 14 are similar views of known filter circuits for comparison purposes; Fig. 15 is a diagram showing plots or curves of image-impedance characteristics of my filter circuits; and Fig. 16 is a similar diagram showing plots or curves of corresponding attenuation characteristics.

In Fig. 2, there are shown two networks, each having only one pair of terminals 1, 1. In'Figs. 1 and 3 to 14, there are shown four-terminal networks, onevpair of the terminalsat 1, 1 or 1, 1f and the other pairat 2,2'or-2, 2'.' One of these two pair may be the input terminals and the other pair the output terminals.

Two networks are said'to bev equivalent when they have the sameexterior efficiency rcharacteristics asfunctions of frequency, irrespective of what might happen interiorly. In the case lof equivalent, two-terminal'networks, all their exterior properties Acan be expressed by the drivingpoint impedance function e. The function z will be `referredvto as the frequency characteristic of a two-terminal network. l Four-terminal networks are determined by three such independentl functions of frequency; or, if'desired, there are four such functions, A, B, C and D, of which three only are independent. Two four-terminal networks are equivalent, therefore, if'theyhave the same frequency-characteristic functions A, B, C and D. A complete set of such functions may be obtained, vfor example, in the following manner:

Letting E1 represent thecomplex'voltage at the input terminals 1, 1; I1 represent the complex current at the input terminals 1, 1; E2 represent the complex voltage at the output terminals 2, 2; and Iz represent the complex current at the output. terminals 2, 2; then there are two linear equations betweenthese voltages and currents, as follows:

A B c DI-l Then A, B, C and D are the said. functions of frequency, and will be referred to as the frequency characteristics of a four-terminal network.

A four-terminal network is said to be symmetrical when its exterior properties remain unchanged upon interchanging the input terminals 1, 1 and the output terminals 2, 2. In the symmetrical case, allv characteristic properties are determined by only two functions of frequency, as

A circuit is a canonical circuit for a certain kind ofi-network if it can be made equivalent to all circuits of thiskind by suitable choice of the size of its elements.

The circuit of Fig. 1 is a canonical, symmetrical, four-terminal network, as is proved in my paper entitled, ber die Variabeln eines passivenVierpole in Sitzungsberichte der Preussischen Akad- -emie den Wissenschaften, 1927, the said two independent frequency characteristics being indicated at zi and z2, respectively, in the pairs of opposite arms of a Wheatstone bridge. These functionsl ai, zare always physically realizable driving-point impedanc'es. In the cases where the resistances are Zero, one 'possible canonical form of a1 and a2 is illustrated in Fig. 2.

The circuit of Fig. 3 is one example of a canonical, unsyrnmetrical, four-terminal network without resistances.

wave filter in series with a tightly-coupled transformer T or a phase-correction network, or both.

It will conduce to clearness not to enter into long-drawn-out.mathematical proofs here, but to state concisely the. general steps of the mathematical treatment, and the results obtained thereby. VSome of the proofs will be found in several of my publications, as follows:

Vierpole, 'Elektrische Nachrichtentechnik, vo1. v1, p. 272, 1929.

Uber eine Klasse von Funcktionen, die die Stieltjesschen -Kettenbrche als Sonderfall Y enthlt, Jahresbericht der deutschen Matheand their product, 212,2, which, naturally, are also rational lfunctions of the `rfrequency J'.r yThe reasons for this will appear from the following considerations:

The propagation constant is ,usually represented by the symbol l', and its real part, which is the attenuation constant, will be represented by A1. I depends on Z2 2 1 only, according to the following equation, in which e represents the Naperian base of logarithms:

Themage'impedance 'Z may be determined by the equation l 4i/Z122 The wave-filter circuit will be most eflicient when theV following conditions are `most nearly met: In the attenuation regions,

. v f Z1 and in the transmission bands,

where R is a constant and equal to the impedance of the given sending or receiving apparatus. It

The circuit of Fig. 3,A by suitable choice of the elements, can be madel equivalent to a combinationf of a symmetrical tenuation in the attenuation bands and zero attenuation in the transmitting bands.' To make a good approximation to zero attenuation in the transmitting bands, it is necessary that the resistances be nearly zero. In the following portion of this specification, the resistances will be wholly neglected for this reason and also because it will simplify the discussion without in reality detracting fromthevgeneralitywof the method.l

In addition to the above requirements for i andaizz for attenuation, it may be Anoted that the requirement z1z2=R2jservesto reduce the echo eiect when the filter is vconnected with a long line having the characteristic impedance R.

There are many different types .of symmetrical wave filters and all of these maybe designed from my formulas; The most simple types may however, be reduced toy four: lband-pass filters; low-and-high-pass filters; low-pass 1ters; and high-pass filters. l f y All possible wave filters of Kall four of these types may be obtained by using the following tables, except that in the case of classes a, b, c

only those classes are represented by the formulas in which the total number of zeros and poles is symmetrical on the two sides of thecriticalv or cut-off frequenciesw-1 and w1, as such classes only may be expected to be of importance, in practice. The frequencies @-1 and w1 may be defined as the frequenciesseparating attenuation from transmitting bands, and may be measuredin radians per'second. It will be understood that w-i and w1 represent particular, cut-off frequency values, and that, in general,

Iw=21rf and y L--iw i representing the imaginary The wsfwith subscripts appear Ain these tables in their naturalorde'r of magnitude from left to right. For example, the resonance or the antiresonance frequencies of the impedances zi and z2 appearing in the tablesfor class 6 are Allthese ws are arbitrary constants.' The only othersymbols appearing in these tables, m and n," represent also arbitrary constants which are positive. In Table II, 4as is identical with c', a4 with d, etc. In Table IV,` a3 is identical with 7, a4 With etc. With further reference to these tables, it will where lbe understood that filters designated according lo for a, low-and-hgh-pass lter.

'I'he following formulas represent:

for a. band-passk'lte'; I

Table! mmm teristic for any of the desired classesl, 2, 3 Table II will similarly'supply the corresponding value of the image-impedance characteristics for any of the desired classes a, b, these (ab) H letters representing any whole number. These same values will correspond to and respectively; o forV a iownd-ngh-'pss niter.;

Tables III and IV'will similarlyrespectivelyfurL nish the values of l1 and y i for a low-pass filter and L and for a high-pass fl1ter.-.Thus, given the product and the quotient, it is possible to compute the quantities themselves, zi andr ."r..`

For example, as already stated, class 3d represents a band-passvrltelgwith. attenuation characteristic class 3 andirnageimpedance characteristic class d. To evaluate this lter, the following value is taken froml Table 1I, for a band-pass lter, for class 3:

Now,

and

I l The two edualexpressions aboveffor Hence, for class 3d,

and

Corresponding to each four-terminal network represented by the classes-thus obtained,there exists an inverse orreciprocal four-terminal network, as defined, forexampla on page 281 of a paper entitled Verpole, in Elektrische Nachrichtentechnik 1501271929. The inverse four-terminal net-work isa net-work for which the frequency characteristics are respectively the reciprocals of the corresponding frequency characteristics forthe original net-Work. Each inverse or re- ',ciprocal lter network is, therefore, a filter of thel same type as that of vwhich it is the reciprocal.

and

and .i i

.. It is not necessary 'to compute tablesfor the z z l will be taken, for class, as the value of From Table II, similarly,` for classd, the following reciprocal is taken;

thus

and the values 'of 21 and z2 will, of course, be found to be the reciprocals of those deducedabove for class 3d.

Asthe absolute value of the vlogarithm of a number is the same .as the absolute value of the logarithm of the reciprocal of the number,

towel# Condensed'expressions of thisv character coveringV a class and its reciprocal will be used in claims 1 to 10 inclusive, and other claims.

As I have shown in myl publications, every pos,- sible symmetricalfour-termnal lter network, whether or not heretofore known in the art, belongs to one or another of these classes. The values of ai and z2 corresponding to ever-y such network may be computed. See, for example, the explanations given above-mentioned paper, entitled Vierpole. Conversely, knowing thel values of e1 and e2 corresponding to any given, network, it is possible tocornpute their product and their ratio, and thus to fix the class to which such network belongs.

It follows, of course, that some of the networks obtained from Tables Ito IV are already known in the art. Most of these networks, however, andparticularly most Aof those obtained from the image-impedance classes,vare original with me. Even in those cases, however, where the networks are already known, except in the very lowest classes, my invention has. a further advantage inv that the arbitrary parameters m, p and the various ws may be` so. chosen as to produce the most efficient results.v Heretofore, these known networks have beendesigned in haphazard fashion, and the results obtained thereby have ldepended more or less on chance. According to my invention, it is possible to compute beforehand those values of `the'parameters that shall yield the maximum elciency in any network, new or old. Old lters', therefore, may be so designed as to `fulfil thecrequirements imposed 0.11 them more accurately than has been possible heretofore. Another advantage of my invention, applicable both to old and to new networks, but particularly advantageous, for present purposes, in connection with the old networks, is that it is possible to reduce the number of circuit elements that may baemployed, thus ,lesseningther cost of production ofv the .network circuits.

With regard, first, to the choice of the best parameters, the labor attendant upon this work may be shortened by graphically using an extensive collection of curves, published in the above Y paper Siebschaltungen. These curves take into account not only the capacities and the inductances, but the 'resistancea as well.- `For simplicity, the method of treatment will be explained in connection with a concrete example, but it will be understood that the method is perfectly general. Parameters dening vresonances and anti-resonances andbeing proportioned to provide a free transmission band are` advantageous if they are spaced inthe attenuation range remote from the band limit or band limits on a progressively closer scale near to the bandlimit tor provide substantially uniform characteristic impedance (imag impedance) throughout the band.

Let it, then, be assumed that it is desired to obtain a1 and z2 with the most satisfactory parameters for a band-pass filter having a narrow transmission band; that is,

Reference may now be had. i9 Eissll .1.5 ild 1 5- which containplots or curvesfof image-imp'eclance,l and, aliieiliiaiin Coriiaim 4W'i1f1-t'h best Daraineters. lin these'figures, thevabscissa is taken, not. the, pulsaiance w, but as the normalized frequency il., where This is` a mere matter:ofconvenience, as' it yintroduces symmetry, and thecut-.off frequencies are determined by l 'ihe of,orcli,.r1siesfortheimage@impedancev or surge-impedance Curves Fg- 15', is' Ibsaiithmisano, is so chosenihai: iii? @Cristianiio which.

e122 approximates is unity..V All cases may easily be reduced-to case by dividinsihe. values 0f 21., .aad necessary -iorfzihe Pariiiilarlier de- Sired by the 1J rev,ifzuslvv given., ,or Knots valli Qi R., of the transmitting or receiving apparatus..` In the following discussion-andalso in the Tschebyscheff formulas hereinafter given, e1 and z2 will be assumed to havebeen so. reduced-so asti.)

have the following values:` Y,

,Fsdarc For filters having comparatively narrow transmission bandsj, it is-possible to chooseA image-impedancej` and attenuation cl,raracteristic'sV that are symmetrical Withrelspect to'bothcut-iirequeii* @es @ed @-1, iria is; wheatr,

- curves c, d, cand J.A *'Ihe ordinate V21?? is on a'logarithmic scale,f. but the abscissa, representing n, is on anatural scale. These ,curvesc, d, e and are sochosensthattthey have in this logarithmicscale' the minimum deection from 7 the straight line v intheinterval.y c

. f-'. f" is called, .-approximaticn linfthe 'Ischebyscheff sense and'iseiectedjby makingequal the deviation at these 'points of maximum deviation. Over ihie raise, niet lilar .case- 6 illustrated, in Figfl ythe'bc'nlndary values for n, defining the range, are +1=0.95 and -x=0.95.

The vboundary value x=0.95 is' shown; on thev The boundary limit k may be taken as close to unity as desired-but it canv notequal unity, this value unity being the limiting cut-off frequency of the transmission band.

As inspection of these curves c, d, e and f will demonstate, their maxima and minima occur alternately between the boundary limits with maximum or minimum at thse umits if the passage through the value of maximum. de-

viation be considered as a maximum or minimum. For example, the curve ,thas three minima and two maxima in the interior of the interval and two maxima at the boundaries. Two of these three minima are indicatedat F1 and F2, the third being a reflection, 'about the straight line 9:0, of the minimum F1. One of the two`inte` rior maxima is shown at F3, the other being similarly a reiiection thereof.A One of the two bounary maxima is indicated at F4, the other being likewise a reflection thereof.

The curvesk oscillate' between two straightlines which are parallel to the line i and both of which are distantfrom this line by the same amount, log H, defined hereinafter, and on both sides of it. For example, for the curve f, the two straight lines are indicated at fr'and fn, where H is the maximum value of the reduced surge impedance over the Tschebyscheman range -x to +r. The deviation log H is the same for all maxima and minima of any one curve.

The distance, log H, decreases andthe number of interior maxima and minima increases, as the class number increases, for the same, given K. For example, curve f has a greater number of maxima and minima than curve e, which, in turn, has a greater number than curve d, and so on. Again, the distance log H, as may be seen by inspection of Fig. 19, is smaller for the curve f than for curve e, and is smaller for curve e than for curve d, and so on. It will thus be seen that the greater the class number and, therefore, the smaller the value of log"H,`the better the approximation to the ideal, limitingrequirement that 2122=1.

If, for a given class number, the limiting frequency x is chosen nearer. and nearer to the cutolf frequency 9:1, then log H approaches nearer and nearer to infinity. Toi'maintain''a given value of log H,-therefore, if x is increased, it is necessary to choose a filter of higher class.

The best parameters foraband-gpass filter, so as to improve the'"attenuatio':n"k andv impedance characteristics, may be chosen?,L according to my invention, wheret'he following two requirements are fulfilled. First, the best parameters for the image or surge impedance, for a given class, are obtained when behaves'inthe manner just described in thefre-V quency interval w w w.

which corresponds to the normalized frequency interval l -lc Q lc before mentioned. Secondly, the best parameters for the attenuation constant A1, for a given class, are obtained by the use of another setof similar curves in two other normalized frequency intervals Y l K1 Sl x2 1 and 1 x3 l x4 The valuesof w corresponding to x1 and K4. may be such that above or below those values, there will be no attenuation requirement.- The value of A1 thus obtained may require certain corrections for resistance'and for proximity to the limiting frequencies, when it is desired to, obtain final values of the attenuation constant. The values of w corresponding to K11 and x4 may be such that above and `below them no requirements regarding the attenuation have to be met. In particular, w., may bev zero and w., may be innnite. The values K2 andx3 may be chosen as near as possible to the cut-off frequencies 9:....1 and respectively, but they can never `become quite equal to these cut-off frequencies.

In Fig. 16 are shown attenuation curves 1, 2, 3, 4, 5, 6 representing for a band-pass filter, but on a particular ordinate scale. The ordinate is the attenuation constant A1, hereinabove defined. To the straight lines log =log H zl of Fig. 15, as for example, the lines f1 and fn, there corresponds in Fig. 16 a line of minimum attenuation A1 mm=constant=4-7 in the pmt the value of K3 approaches more and more to the cut-off frequency Q=l.` 'I'he value of x3 is shown in Fig. 16 by a small circle on the axis of absciss at its value for class. A

It will be understood that the full attenuation curves are not shown in Fig. 16, because the portions of the curves to the left of Q=1 are reflecriesame SiOI-IS 0f the portions shown, and-therefis no attenuation in the interval f.

'YCorresponding con-sid'eiia'tioris` appk to-'others types of filters than theband-pass filter. For eX- ample, the curves of Fig.; 1-5` may be the N/ curves for a low-and-high pass-Jiilvter.l

changedto.. .l

curves for fzz; I' N4 will be obtained fora low-and-high-passy.filter. In such cases," the straight Aline Y or their corresponding curves, behav'es "'just' 'described, Vthey will be saiduto exhibit a`vT s cheby chei behavior, and the said parameters will'l then be known as Tschebysche parameters. .The various minimaol'y attenuation overja prescribed part or parts of theattenuationregin are vsubstantially equal, and theA` various maxima and minima of surge impedance over a prescribed part or parts of the pass region al1 deviate to substantially the same extent from a predetermined value.V The values of the .Tschebyscheffrparameters depend upon the frequencies of the iirst and the last maxima and minima (JK,K1). The intervals K to K1 and so on will be referred to as Tschebyscheff intervals. l f i' 1 It is to be understoodV that the Tschebyscheff parameters are not unique; but that there may be diierent sets of such parameters, corresponding to different values of K or log flgyfqlf a givenfilter class. The choice of lclass and parameter is suggested by the practical requirements for the image impedance andv for the working attenuation A '.(it isgtobe noted that this A is not the saine A Aas theA heretofore employed inthe equationfor Ei') .j;f,This working attenuation is defined as and where J denotes the current ow in the receiving apparatus when it is'directly or immediately connected with the sending apparatu's' andwhere-J 1 is the corresponding current when the wave lter is interposed between the sending and the receiving apparatus. I

As I demonstrate in my above-mentionedrpaper on Siebschaltungen, V. D. I. Verlag, Berlin, 1931, the working or total attenuation is constituted of two superposd.Y parts, A1 and A2, together with a correction -v 21 r-l-z-l-Aa A2', similarly, 1s afunction.pf I. I

The ladder-type lte chainof/Fig. stitutsdf-` of simple, equa1f.911r:termina1 networks @sections sechsectien.consistinsofa vSeries c011- denser lll-and.a.-parallel-connected condenser 12 and inductance lf3 lin; shunt,. Each such section belongs to classflqag` The iiltenchainof Fig. 6 is constituted of sectionsthat differ more or less from one another. These sections are connected togetherand arranged I according to the description by OttoJ. Zobeland is known as his M-type, with m -".0.6.. 'Cm'pa'rejhis paper on Theory and designer-uniform and-composite electric wave lilters,2 inthe ell System; Technical Journal, vol. II, Noi= l; 'J anuary,`192,3,.lpage 30. This filter belongs to my class dhrA: number of variations of such special networks or classes of networks have beenf'proposedandusedin the prior art for the selection vand""propagation 'of 'prescribed CII ranges of frequency. v"Itwill now be understood 'y that my invention is an improvement over these old circuits in severalj particulars, among them, that I am able to meet demands forel-.which the prior-art circuits were not adapted, and even in those cases where they Were more or less satisfactory, I am able to provide.betterparameers so as to render these oldI types of circuitsvir'iore efficient. l

In order the better to understand how the parameters ofthese filters may be improved in accordance with the present -invention, let the number-:Lof sections of the iiltergoffFig,` 5 be represented by the numberm ""Thengthejlter has an attenuation class u. The corresponding attenuatqgonstelntsj A1 for eachA 'of the six classes,

respectively, of Fig. 16,`wliich show, in eachnase, the `vfir-stbranch-.only ofthe curve..r Curve 6 of Fig'fl, similarlyfisthe'attenuation curve of' the filter shown in Fig. 6. It requires 'no more than ,Tscheb'ysckieirpamamtersszFor example, I a lter of my class 4, .with thezbestparameters, will give. the same results as, or better than, those obtained with the circuit of Fig. 6 and with poorer parameters. 'Ihis is for the attenuation requirement In the curves ofFigs, A and 16, the ohmic resistances, for simplicity, have been neglected, but it is possible to correct the curves for such resistances, so that the method of the present invention is perfectly general*I v According to present-day practice, the image impedances of knownfilters are designed with the aid of end circuits. ,A',serious disadvantage of this practice, if a good image-impedance characteristic is desired, is that the attenuation is unnecessarily partlydetermined thereby, with consequent loss "of efiency.' Tlius,ainecessary consequence of the designofZobels M -type (m=0.6) circuit, Fig. 6, is that frequency'values 'of infinite attenuation are In designing filter circuitsaccording to my invention, on the other hand; the attenuation and the image-impedance characteristics may be chosen quite independently of one another, and their values are not dependent'upon the presence of any particular, single parts of the circuit, such as the end circuits." y

' All Tschebyscheff parameters may be computed by formulas mathematically derived in my paper, entitled Ein Interpolationsproblem mit Funktionen mitpositiven Realteil" in the Mathematische Zeitschrift, 1933, as follows:

when p is odd when p is even.

2V Sn (P, q being whole numbers, q p) the normalized resonance and antiresonance 4frequency parameters. i l

The minimum value of the attenuationl constant in an interval of Tschebyscheff behavior is sn and Kmay be calculated in any well known manner, as with themaid of the' formulas and tables in Funktionentafeln, by Jahnke and Einde.

The value Aimm exists'at certain special frequencies only, and the above Formulas (I) and (II) express` the following'law for those special frequencies: The various equal "minima of Vthe attenuation occur at those normalized frequencies that are equal to the ratio between the normalized limits of the Tschebyscheian intervals and the normalized resonance and anti-resonance frequencies of the attenuation function. A similar law holds good for the position of the various minima and maxima of the image (surge) impedance. y

Using the above Formulas (I) and (II), the values ofci and zz for 'the iilterin question may be obtained by suitable transformations. Thus, for a low-pass lteri'n the Tschebyscheff interval A wx1 wlr wxg 'V w1 o1k1 wllq the Tschebyscheff parameters for the classes may be obtained as follows from'Equations (I) Vbyscheff parameters for the same classes may be obtained from the following equations:

V2m=y1 when p is odd, and 'Y 1 i l i# iwf?, when p is even. n

In both cases l .l :wmf-wow l u (witg-(02h01: n. and

, and

= low-and-high-pas's lter 'classes ml L '-5 It is necessary tobbtnathe Tschebysc Heffk parameters for the lter classes correspondi'ng v.to A 'Y l-lTl/i When p is odd, and y #wifi whenpis even. For the corresponding high-passrjltr-in Ithe same Tschebysche finteryal, the equations to be used are when p is odd, and

" gli when 21 is`v even.

Here,

' tained inthe above-discussed speciajl'casefforthe low-pass (or high-pass) `cltasses 1, 2, 3,

Formulas somewhavsirnilar to these may be obtained, by suitable transformations-,from vthe same Formulas (I), for band-pass lter and for and A when p is odd,

i -when p is odd, and

approaches infinity, *airnstill retaining when p is even; I and for a low-andl ,A Tschebyscheff iniaerveielgand if where and much simpler: form; ,f

@fea-n* p=1. 2, 3 L u" I in the Tschebyscheff interval corresponding to 5 the followmg, s1mple, is obtained: (I) are appropriate.

@ -weeg); l' i f. ',fzlfFfyl' (ira-1). when@ is odd', and

i.f'reqiiericy transformation Fil the following formla'sfobtained fromFormulas 15' 20 To obtain the Tschebysoheff parameters jforv`l #g1-,Fl 204" the band-pass lltersand thejlow-and-high-pass Y1 1 the c1 Ssesw; when p is even. ters for a K The corresponding formulas for low-and-high- As an illustration, for class b, y

where Finally, the Tschebyscheff parameters for pand-pass filters, (and correspondingly for lowandvhigh-pass-lters, as will now be understood) 60 of classes corresponding to 5w .A l .pri-'1*'3'5 'A 65 may be approximately obtained, for narrow whence bands, that means for from Formulas (I I) by the substitution so that be derived:

is the value given for class e in Table II, multiplied byl For class g,

VZV-2' is the value given for class g in Table II, multipliedby f i [E @+Qu (w-I-w-Xw-i'wXw-i-w-v .ww-1 @Meow-w+@ andsoon. Y' For narrow bands, the quantities in the brackets of these formulas may, for practical purposes, and without substantial'loss of accuracy, baconsidered as constants incertain'neighborhood lf thebands. 1 l, As is demonstrated by the mathematical theory developed in my saidpape'r in the'Mathematische Zeitschrift, it is, possible to use' ,also other very favorable parameters, though not nec-Y essarily Tschebyschef lparameters particularly in the design of more complicated lters, ,and more especially lters with more than two bands.k Theseare not enumerated in detailherein, mere-l ly in order to save space, and it is thereforede-i sired that the mathematical treatment giyerrinv the said paper be considered as embodied hreinj and considered a part hereof. Such complicated lters may be distinguishedby thelnumberkof points in which the characteristic curve 'corre-A sponding to (or, as the case may be, to

is intersected in the most favorable .case-by a parallel to the abscissav axis of'frequencyin the transmission (or, as before stated, the attenua-f tion) bandsn Forexample, in Fig. 15,.the.ycurve cZ-.willrfb intersected'fbysuoh 'a horizontal line, in

the. transmissionfband, at' four points, two of which'will `be-visiblein Fig. 15, and the other two f are" the reflections thereof.' The multi-band n1- ters, inA accordance with the present invention, are such that it is possible so to choose the said horizontal line. that it :shall intersect the charstant aspossible in the' transmission bands, but this haslnever heretofore been possible of attainment except Ito a "veryrough degree of apl proximation. lAccording Gtov my invention it is possible to attain this result toany desired degree.

of'accuracy. i

One advantage .of the useof 'Ischebyscheff pa-- rameters is'the .possibilityofireducing the classication number of a. lter circuit fthat has 'tosatisfy givenattenuationv and surge-impedance reo1uirements.lgY

-" Using..thesemethoda` it is possibleto choose the number, size and classes of the elements, both capacitative and inductive, so as to yield the most desirable results, with afsmaller number of circuit elements, as described more fully hereinafter. Usually,ffthe=emphasis is placed upon the number and sizeof the inductanoes, as they are the more exp'ensivefelementa` but it will be'understood that economy .is equally possible in connection with the-capacities. H l y It has 'now been 'shown how to obtain the functions 21 and afg to be used, Yfor example, in the required tercircuit of Fig. ALand how t'o obtain,

also, the best parameters f or such functions. It should now be explained, however, that 'corresponding to each set of functions e1 and z2 thusl obtained, there are manydifferent, though electrically equivalent, filter circuits. understood by persons vskilled in the art without further description, though reference may be made to a paper by R. M. Foster in the Bell System vTechnical Journal, vol.v 3,- Aprill, 1924, as well asto my paper in the Archiv fr-Elektrotechnik for 1926. In my said paper in the Mathematische` Annalen, aswell as in my said paper in th ElektrischeNachrichtentechnik, voigvr, 1929,.

" at page 272, indeed, itis explained how to construct all circuits corresponding and equivalent to any given network. If the frequency characteristics of a filter be expressed in algebraic form, the corresponding lter can be constructed in knownrnanner iromthe partial or continued frac-- tions er' and Izzi- S To each choice of these functions "there corresponds a large number of 'equivalent circuits, several examples ofwhich are illustrated inthe drawings. f

One such circuit, as will now be understood, is illustrated in Fig. 1, assuming that e1 and e2 are of the `form illustrated in Fig. 2.

The impedance ,er of Fig. 2, may be constituted 3 and 5 are indicated as having? the values Li and L2, Ls, Lr, respectively, and, the condensers 4 and 6 the reciprocal valueslDl, D1,

D3 2 similarly constituted of a condenser 1, having a. reciprocal. value DH1, in series with an inductance 8, having a `value`Lr+1 and with groups: of simi.-

larly parallelearranged groups of inductances and capacities 9, 10, having inductance values Las, Lr+a Lm'and reciprocal capacity values'DH-i and Dr+2, Data. Dn, respectively.

The filter network` of Fig..4like.that of Fig. 1, has two inputL terminals 1, 1` and twooutput terminals 2, 2. The .indicated values of inductancc and reciprocal capacities are the same as in. Fig. 2. Between the left-hand terminals 1, 2, as viewed in Fig. 4, there are connected, in series, two capaci'.- ties Il having reciprocal values Df+1. At a point between these two condensers, there is connected a further'capacity 12 having a reciprocal value is connected in series withv the capacity l2..v Fur-J ther connected in series are a number of parallelconnected groups of inductances 1.4 and capacities having values L2 D2. Lf

; These areconnected in series to the. other terminal 1 through a series connected inductance 16 having a value L1+1 T The coils markedV LH-zn, 13. Lr+i 2` 2 2. are individually tightly coupled to other coilsA 17, 18, 19, as illustrated, so as to produce a one-Vto-one ratio in the resulting transformers, these coils 1'7, 1 18, 19 being connected in se'- ries to a point between the coils; marked 2 and 2 and to the other terminal 2." The ratio is marked on the drawings 1:1. 'This is `to explainthat the points and 21,*for example have not the same voltage. If they had had the same voltage, the ratio could have been 1 :1` and thecoil 1T mightr have been omitted. y

The network of Fig. 4 comprises two circuitsr one, from the iirst input terminal 1, through the? upper condenser Il, the condenser I2, the. coil 13, all the coils 14, and the "coil 16, tcr` the sec-r ond input terminal l; the other, from the third or output terminal 2, through the lower condenser 11, the condenser 12, the coils- 14 and the coils' 17, v18 and 19, tothe fourthlo-r output terminal. 2.'.

These two circuits are inductively cou-pled positively to the left of the said` point betweenv the .Da respectively. The impedance z2 of Fig.

common, but to the "right of the said` point',` the'j circuits are separate.

It will be understood that the invention crom-l prises all cases of lter networks derived accorde ing to my formulas'andhaving one circuit through the input terminalsand another circuit through the output terminals, the circuits being induc-` tively coupled to one side and to the other side of an intermediate point, and an inductive coupling to one side of theintermediate point being negative relative to an inductive coupling to the other side of the intermediate point. The coupling may, in particular," be realized by common coils, or by tight couplings or by a combination of th'e same'.

It may be shown that the circuitdiagram of Fig. 4 represents a circuit equivalentto the circuit obtained by replacing ai and z2 from Fig. 2 in Fig. 1. This may be demonstrated mathematically, using a linear, afiine transformation, as described in my. said paper in theJahresbericht der deutsches ll/Iathematikervereinigung.v and ,in the said paper in the Mathematische Annalen. It

will be made obvious also, by calculatingthe` open-circuit andthe short-circuit impedances in both figures. For-example, iftheterminals 2, 2 be regarded as open-*circuited both in Figs. l and 4, then the impedance across the terminals 1,1inFig.1is H Il tzf as is obvious on mere inspection; andthe same is true of Fig. 4. I

It has now been shown-how toobtain any desired lter network, old or new, and howto evaluate the best parameters therefor. A'further feature of the invention was stated above to reside in the reductionof thenumber of circuit elementsl that need be employed in such circuits, thus re ducing their cost of manufacture. Y

The iilter circuitr'represented in Fig. 4, which may be termed the.L canonical circuit ofwhich all circuits of the corresponding class are equivalents, has, in general, the minimum number of capacitative and inductive. elements.

Mention has been made fabove of thefact that the invention is not restrictedtof the design of symmetrical filter circuits, and one unsymmetrical circuit in accordance with my invention is illustrated dagrammatically in Fig. .3. In my said paper in the Mathematische Annalen, 1931, Ihave demonstrated' that this is a. canonical form of circuit which by suitable dimensions and choices of itselements, may be made equivalent to anyl unsym'metrical.four-terminal network' with negligible resistances. f This canonical form of ,circuit comprises a number of parallel-con#- nected coils andcondensers 24,r arranged in series with a coil 26 and a condenser 27, the lastnamed condenser 27 having a shunt-arranged coil 2'8 and condenser 29.. An idealtransformer T is connected in this shunt circuit.- By"ideal is meant a tightly-coupled transformen having great inductance. The arrows connect coils that are mutually inductively related, and only the coils so indicated by the `arrows are so related.l

This canonical circuit is equivalent to a symmetrical wave-filter circuit connected in series vertible into, and equivalent to, the `circuit of Fis; 4'. Y

In order to save circumlocution of language in inclass Bb'. Y 'I'he Acoils. havingl the same' lettersfare 'on a single core,with the-'result V-that,for manu'- facturingpurposes;theyvmay be considered as a single coil, thusreducing thef'numberof coils required by the .lteni It is alwayspossibleyin lter circuits of the type illustrated i'n Fi'gfl,` to place -twoV opposite,symmetricallydisposed coils-"on one core in this manner.; Suchl circuits are desirable in cases of line balance'where,1for'examplefthe unbalanced equivalentcircuit of Fig. 8 could not be directly' used, except-with further inductive coupling Aand a change of connections. Fig. 8, of course, is a special case, of the more general case illustrated in Fig. 4, class 3b.

Fig. illustrates a lattice network having' four branches. The branches 1, 2` and, -the branches 1',- 2 are equal .to-eachIl other, as also are V the AbranchesV 1, '2' and 1', 2.' The brf'an'ches 1, 2 and? 1'-2- each contains acondenser 53aninductance a and an inductance 'b in parallel with a condenser 54. vThe branches 1, ,2f, ,and 1', 2 similarly each have a condenser 55,'an inductance c and an inductance d .inparallell with amcondenser.

AThe coils a, a are mountedfin' asingle core. The

same is true 4of .the coils' 6,11-, aswell as. of the lcoils 'cj Vc and .the coils. d, "d, This lattice-type network is usefulas a@.-wavef-lterfcircuit,of the class 3b* if the'element's a, b, c, d, 53,54', 55and 56 are suitably chosen. FigQllanequivalent circuit to that of Fig.10 and is a special case of Fig. 4. Y

Fig. l2 illustrates'ia imultiple-inductance wave lter embodying the present invention, which -can be made equivalent to the filter of Fig. 10 by suitable choice of dimensionsr'of the4 elementsfand which contains a smaller number of apacitiesand also a more suitable choice of dimensions of the CapaCitiesthan is true-ofthe circuit, cfFig. l10. This circuit comprisesfourbranches, as Fig. 10. The rst branch containsa-condenser 5'7,V and two-coils 58, 59 in series betWeen'the-terminals V1, 2L The second branch,. betwe en the terminals 1, 2', contains a condenser60`, and two `coils 61 and 6 2 in series. The thirdbranchcontains a condenser 63 andtwo coils 64,k 6 5 ,between the terminals 1', 2. The fourth branch lcontains a condenser 66 and two coils 67 and 68'between the terminals 1', 2.-

The coils 58 and 64 are tightly coupled and mounted upon a single core 69..The.same is-true of the coils 61 and 67, up'o'n'a'is'ingle core 70. The coils 59 and 65 are similarly mounted upona single core 71 Witha-third coil 72,'gwhich1atter is shunted by acondenser 73.9 It willbe noted that the coils 59, 65 and 72 are none of them' directly galvanically connected, but are vmutually tightly coupled. AV similar construction" obtains 'for l the shunted bya condenser- "15. The coilsf62,.68 and 74 are tightly'coupled and 'are -allmounted upon a single core '76 similarly to the coils 59,V 65 and .44 is connected between the terminals 1, '1'. condenser 45 is connected between the terminals 2,- 2211A condenser 46 is' connected anywhere be- `lin some. cases, accordingto the present inven- ,tiQL two of the .three mutually inductively related coils maybe g'alvan'icallyfconnected together. In allcases -furthermore,lthe principle of the inven- `tion is unaffected if a direct galvanic' lconnection is unaccompanied by achange of current, as such wouldnot be an-essen'tialg'alvanic connection. It is to be understood,` therefore, that the expression ldirect galvanic connection, or its equivalent, as used, in the specification and claims herein,vhas

` referer; ce to anessential connection, and not toa connectionithat,ishot essential.

The six condensers 5'7, 60, 63, 6 6, ,'73 and! 75 aresmaller in number than the eight condensers 53, 54, 55, and l56 `of Fig. y10.r Av further advantage of y-the circuit `of Fig. 12 isv to avoid toogreat Vcagpacities such as would be necessary, in practice, with the circuit of Fig. 1 0, The dimensions of the elements indicated in Fig. 12 were calculated beforehand for av given attenuation requirement and were actuallychecked experimentally.

Fig. 13 is a diagram illustrating a known, chaintype lter, with `three equal sections, the Whole lter beingof class 3 bf", each section, as shown Vin 'Fi`g.'14, being'roi class'llf; Each branch contains a'series combination` of capacity 30 and in'ductance '31.' "'It is ,always .possible to make the circuits of Figs."10,'11"and 12 equivalent to the wave filter of Fig. 13 by suitable size of the elements', but the'converse isi not true. The circuits vof Figs.' 10,'11 and 12 can be designed with better frequency characteristics than is the case withFigi 13, using the Tschebysche'ff parameters. Inrthe circuit. of Fig.'` 13,.,there are six coils and twelve condensers, counting oppositely-disposed coils as single coils,.as before described. The circuits of Figs. 10, 11 and 12 each have four coils, and eight condensers, ve condensers and ,six condensers, respectively. L

. 'I he ,circuit of Fig. 9'b'elongs to `Vc:la,ss' .v3b and is equivalent to the circuits of Figs. 7 and 8.

`-The circuit elements of- Fig.'9', therefore.' are represented by the same symbols as in Fig..7. The coils 35,'36 andf3'7 are tightly connected together.

The coil 35 is connected across the terminals 1,

1' in 'series with a coil 38 `and a condenser l"39.

43 are connected in series between the terminals 1 and 2 by the same conductor 40. A condenser A tween the terminals 1 and 2. An inspection of thecircuit of Fig. -9 will show that itcontains a fewer number of elements than in the 'circuit'of Fig. '7. vCounting theccils 35,

`Y36 and 37 as one element, as before described, the rcircuitof Fig.` 9 contains fourJ coils :and ve condensers. The circuit ofFig. '7, on the other hand, hasffour coils and eight condensers.

Tlieequivalencef of these two circuits may be demonstrated, for example, by calculating the 'taining such reciprocal isv described. more `at length in my copendingapplication on'articial networks. Serial No. 499,233, led December 1,

'1930 which has become S.1Patent V1i-'1,958,742

May 15, 1934.

equals 1;*9-89 The knovvn Aoi? contains sin coils '3b*. One equivirlerit"circuit', therefore, contains L -coils"and 5 condensersjas demonstrated in It is obviously impossible to describe in detail all 'the special applications of my invention. They will be understood by persons skillled in the art' Withoutjfurther de`scription".v It' is therefore desiredfthat' the appended *claims `b'e broadly construed, uri-limitfed;A except' in'sofarias limitations may be necessary tobej imposed by thestateof l. A four-terminal, -lotvland-high-pass, filter network having frequency characteristics 21 and `z2, where A v Y [-I z1272 where m is :an arbitrary, 'positive constant; A is the imaginary, frequency parameter; n-1, and w1 are cut-off frequencies;- and i5-a and m. are arbitraryy resonant and. anti-resonant frequency parametersin the attenuation band.`

2. Krom-terminal, I owand-high-pass. filter network of imageimpedance class 21h-1 or tvhere 'n is any Whole, positivenumber greater vthan 2; m is anarbitrary-,positive constant; A is the imaginary, frequency parameter; 1 and u yarecut-off. frequencies; and a1 etc. arev arbitrary resonantVl andfanti-resonant Afrequency parameters inthe attenuation band.

3.A four-terminal, lowandfhigh-pass, filter network of image-impedance `class 2n or 2n* having frequency characteristicszl and 2 2, where 10er/F21 equals e. l (2+wa,12)0\2+wa,a2) (X24-warrig) l where n is anylwhole, positive number.l greater than 2,; mr isan arbitrary, positive constant; x is the imaginarmfrequency parameter; w-i and w1 `are cut-off frequencies;,and ma etc. are, arbitrary resonant and anti-resonant frequency paramef ters in the attenuationband..`

4. A .four-terminal-bandf-pass, filter network l having frequency oharacteristicsgzi and 2:, where flaca/5,21 if,

., oww-awww# where ,u is an arbitrary, positive constant;v x is the imaginary, frequency parameter.; w-i and w1 are cut-of frequencies;' anda-a andw are arbitrary resonant .L and j 'anti-resonant frequency parameters in the attenuation bands.. .1

t. rmfhiaimanpage,aiterfnetwbrk of 'and'twelve'condensersIt belongs' tomy class 'imagefirnpedancclafss dg 1 df. @echarme-fre- 'd :ery-here'- quency characteristicserand zz Whe re.` equalsV 4 last factor of :the denominator isfomitted,` for n even; n is any 'iis/I iole,- p sitivel number greater than l2; p. i'sanrarbitnary, positive constantfr is the imaginary.,frequencyljparaneter; 'a5-1 'and tu are cut-off' frequencies; 'and' 1-u, etc. are; arbrary resonant -and anti-resonant' frequency parameters inthe attenuation bands.

7. A four-terminal;high-pass, filter network'of image-impedance class Znf' or `2n1* having frequencycharacteristics@ and ai; Where equals Maat-e owne) m where n vis IanyWhole?'positive number greater than 2;' m isjann arbitrary, positive constant.; ik is the imaginary;v frequgiericyy parameterj' au" is kthe lcut-off frequencyandnazp-z etc.l are arbitrary in the attenuaticn pand. i

8. A four-terminal; high-pass, fi1ter network of image-impedance class 21`v `or 2n* having frequency characteristics zi and zg; where I f or its reciprocalequals resonant and ant'iirsonant'frequencyparameters log m Where n isany whole.. positive number greater than unity; m is an=arbitrary, positive constant; A is the imaginary,v frequency parameter; w1 islthe cut-offfrequency;` and maza-1 etc. arearbitrary resonant and anti-resonantfrequency .parameters in the attenuation band.:

9. A four-terminal, low pass, filter network `of 'transmitting band. f 13. InV awave transmission network comprising image-impedance' class 22;:1 orlagma" having frequency characteristics eiand ez, where where n is any whole, positive number greater than. 2; u is an arbitrary, positive constant; A is .the imaginary, frequencyparameter; u lis .the

cut-off frequency; and w1`etc. are arbitraryresonantl andanti-resonant frequency parameters in `Vthe attenuation band.

where nais vany. whole, positive number greater Vthan unity; .u is an arbitrary, positive constant;

x is the imaginary, frequency parameter;r w1 is the cut-off frequency; and weer etc-are arbitrary resonant and anti-resonant'frequency parameters in the attenuation band. v

11. A symmetricahfour-terminal networkcomprising four branches connected inthe form of a lattice, each branch having a condenser andv two coils in series, the coils of the 'branches being corr'espondingly dispo-sed,` four "corresponding coils being tightly coupled in two pairs, the other four corresponding coils being. also tightly coupled in pairs, two additional coils Aveach tightly coupled to one of the last-named pairs, and two condensers each connected with 'one of the Atwo Vradditional coils. A f

v12. A ,four-terminal, image-impedance. I'llter networkwith more thantwo bands of free transmission having an image-impedance, frequencycharacteristic curve that, in the most favorable case, is intersected by a paralleltoythe frequency axis of ,coordinates in` at least five points in every a plurality' of* vimpedances."two 'impedances adapted to determinethe transmission characterist-ics of thenet'work, said impedances'eacl having a plurality of frequency parametersfdening resonances and anti-resonances and fbei'ng pro'- portioned toi-provide a'free transmissionband or free transmission bandsiand attenuation at frequencies outside.' the transmission band orbands,

said frequency parametersbeing lsfpacedlin the attenuation "range or ranges remote from the log m equals Y. vband limit or` band limits o -n a! progressively closer "v Y scale near` tothe band limit to provide-asubstany equals `were m is a positive constant; i is the imaginary,

' .frequency parameter; u rand w1 are vcut-off fre- Fquencies; and Q and w., are resonant and antiresonant frequency .parameters in the transmission band, m, ma., andrai.. all being Tschebyscheif parameters. l.

CFI

lsAnetwork as defined in claim 2 where m, I

wm, wat etc..are all Tschebyscheff parameters.

1'?. i A four-terminal, band-pass, filter network of attenuation class `21t-1 or.27LA-1". having frequency characteristics, a1 .and z2y whe-re XL1-w-1202+i0m2 02+@1.42) ".(z-Fwazn-kzz) (2'i'wa.12)(}\2+wa.a2) (z-l-wmzn-szM/--wiz where n is any whole, positive, number greater than 2; ,m is a'positive constant; x is the imaginary, frequency parameter; wl and w1 are cutoi frequencies; and wad etc. are resonant and anti-resonant frequencylparameters in the transmission band, m, was etc., all -being Tschbyschei parameters. y V f a 18.A network. as defined lin claim 3 where m,

equals aan etc. are all Tschebyscheif parameters wheren A is, however, any whole number greater than 1.

.19. vAfour.-terrninal band-pass, filter of attenuation. class 2n or 2n* vhaving frequency charact'eristics e1 andfz, where resonant frequency parameters in the transmis- 20. A" network asA deiinedrin claim 4, where jb, au anda...` are all ,Tschebyscheff parameters.

' *21. A four-terminal,vgvlow-and-high-pass, lter 'network vhaving frequency characteristics zi and where "his a positive .constanmjk is the imaginary.

2,3, A fourfterminalf .lQwfand-highfpass.. alter 2; .u is a positive Avconst nosas@ network of attenuation class azn-i or azn-rf'hav.- itive1ponstant;,),isftheimaginary;.frequency paing frequency characteristics e1 and z2, where equals rameter u :is,ithe;.1cut1 .foi fv frequency :fand vmam-1 etc. are resonant andantiresonant frequency parameters in the transmission band, m, @ain-.1 etc. al1 being Tschebyscheff parameters.

aan netwerk? as idenned claim k9 where "log where 11:11. f or `11odd;V vin-F12 word, :o and the -last factor of the ''ieneminator.V is o'rni-tted'for n even; n is any whole, positive number greater than `}\-'"is'the imaginary, frequency parametergb nd aii--are'cut-off frequencies@ Y'and' 1 etc. ar'e'eresonant vand Var-lti- 'resonant frequency lpararneters in the' transmission bands; 11,205, i f-etcff allf being? Tschebyscheif parameters.

24. A network asjdened; in claim 6 where ,nw- 1 etc. are allgTscliebysche parameters, Awhere n is, however, any whole number greater than 1. Y

25. A four-terminal, low-and-high-.pass, Ifilter network of attenuation. classlja2f or "awha'vng frequency characteristics 21 '-andzziwhere baylletc; are'all Tschebyscheif parameters, where n'i sh /jwr n ,than may:

"I 31'A` four-terminal; high-pass, filter network ,of` attenuationclass 2a-1 or atan-1* havingfrequency characteristics a1' 'and' 22, wherey I #+-@Zowat w+ w..,2..22 where 'is any wholefp'ositive number greater than unity; [i is a positive constant; A is the imag- -inary, frequency parameter; w1 is the cut-off freequals '10g IL t quency; and wai etc. are resonant and anti-resonant frequency' parameters in the! transmission bandyp, mi' vete.` all beingI Tschebyscheff pa- 'lastfactor of the denominator is y"omitted for :n even;

n is any whole,"positive number greater than` 1*; a is a positive constant; A'isthe imaginary; frequency parameterbi Vand` glare cut-'off frequencies; and ca .,gl etc. are resonant alndnanti'- resonant frequency parameters the transmission bands, u, o o., 1; etc all being Tschebysche parameters.

26. A network as defined in claim 7 where m, @Mp2 etc. are all Tschebyscheff parameters, 41where n is, however, vvany whole positive number greater than unity?. Q 'Y f 27. A Yfour-terminal, low-pass, filter network of attenuationclass 21u-1 orY 211-1* having frequency characteristics v21 and 22. where'` equals in claim 8 wherein Y is, however, any whole positive number and where m, waan-1, etc. are all Tscheb'yscheff parametersi 29. A four-terminal,` low-pass, filter net-work of attenuation class 2n or 2n*. having frequency characteristics 21 and 22,' where equals ',log m characteristics :mitting bandi 32.V Anetwqrklasdeflned where n is, howevenjany whole .positiyenumber and where u, wa, 1 etc. are .all `."llschebyscheif parameters.

of attenuation ss puilcr= yuw havingvfrequen'c'y i'andaawhere wheren 'is any-whole; positive number; ,L is a positive constantf il is? the imaginary, frequency parameter; m' is the'cut-'oif-frequency; and man etcQare arbitrary resonant and anti-resonant frequencyparametersin the transmission band, um "etcallfbeingTschebyscheff parameters. f

34. A four-terminal, image-impedance, filter network with two attenuation and two transmission bands, having-Av aniimage impedance, frefluency-characteristic,curve thahi-nthe most favorable*.ease,IisIk intersected by a parallel to the frequency axis, of" coordinateslin at least five points din every transmitting band.

35. A Affourf-ternflir'ial, band-pass, filter network 'having' yan1 image-impedance, frequency-charact'eristicfcurve'lthat, inthe'lmostfavorable case, is intersected by' ajjp'a'rallel tothe' frequency axis of Ycoordinates in atE leastfv" V points l in the trans- 36. A. four;terfmnajrk Vfowlpass., alter network having anfimagef-impedance; frequency-characteri'stic lcurve-that, inthe most favorable case, is

intersected by a'parallelto"theffrequency Aaxis of coordinates Ein# at least-'four points in' the transnuttingbandw Y 37. A four-terminalg'hig'h-pass,lter network having van image-impedance, frequency-characteristic curve that, in the most favorable case, is intersected by aparallel to the frequency axis of coordinates in at least four points in the transmitting band. 1

ever; "any whole positive number greater fingir-pass' lter network 

